Controlling the Rates of a Chain of Harmonic Oscillators with a Point Langevin Thermostat

We consider the control problem for an infinite chain of coupled harmonic oscillators with a Langevin thermostat at the origin. We study the effect of two types of open-loop boundary controls, impulsive control and linear memory-feedback control, in the high frequency limit. We investigate their act...

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Bibliographic Details
Published inarXiv.org
Main Authors Hannani, Amirali, Tran, Minh-Binh, Phung, Minh Nhat, Trélat, Emmanuel
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 14.03.2024
Subjects
Boundary control
Feedback control
Harmonic oscillators
Thermostats
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Summary:We consider the control problem for an infinite chain of coupled harmonic oscillators with a Langevin thermostat at the origin. We study the effect of two types of open-loop boundary controls, impulsive control and linear memory-feedback control, in the high frequency limit. We investigate their action on the reflection-transmission coefficients for the wave energy for the scattering of the thermostat. Our study shows that impulsive boundary controls have no impact on the rates and are thus not appropriate to act on the system, despite their physical meaning and relevance. In contrast, the second kind of control that we propose, which is less standard and uses the past of the state solution of the system, is adequate and relevant. We prove that any triple of rates satisfying appropriate assumptions is asymptotically reachable thanks to linear memory-feedback controls that we design explicitly.
ISSN:2331-8422